**Memoirs of the American Mathematical Society**

2004;
178 pp;
Softcover

MSC: Primary 46; 47;
Secondary 43

Print ISBN: 978-0-8218-3491-6

Product Code: MEMO/168/797

List Price: $73.00

AMS Member Price: $43.80

MAA member Price: $65.70

**Electronic ISBN: 978-1-4704-0395-9
Product Code: MEMO/168/797.E**

List Price: $73.00

AMS Member Price: $43.80

MAA member Price: $65.70

# Representation Theory and Numerical AF-Invariants: The Representations and Centralizers of Certain States on \(\mathcal{O}_{d}\)

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*Ola Bratteli; Palle E. T. Jorgensen; Vasyl’ Ostrovs’kyĭ*

Let \(\mathcal{O}_{d}\) be the Cuntz algebra on generators \(S_{1},\dots,S_{d}\), \(2\leq d<\infty\). Let \(\mathcal{D}_{d}\subset\mathcal{O}_{d}\) be the abelian subalgebra generated by monomials \(S_{\alpha_{{}}}^{{}}S_{\alpha_{{}} }^{\ast}=S_{\alpha_{1}}^{{}}\cdots S_{\alpha_{k}}^{{}}S_{\alpha_{k}}^{\ast }\cdots S_{\alpha_{1}}^{\ast}\) where \(\alpha=\left(\alpha_{1}\dots\alpha _{k}\right)\) ranges over all multi-indices formed from \(\left\{ 1,\dots,d\right\}\). In any representation of \(\mathcal{O}_{d}\), \(\mathcal{D}_{d}\) may be simultaneously diagonalized. Using \(S_{i}^{{}}\left( S_{\alpha}^{{}}S_{\alpha}^{\ast}\right) =\left( S_{i\alpha}^{{}}S_{i\alpha }^{\ast}\right) S_{i}^{{}}\), we show that the operators \(S_{i}\) from a general representation of \(\mathcal{O}_{d}\) may be expressed directly in terms of the spectral representation of \(\mathcal{D}_{d}\). We use this in describing a class of type \(\mathrm{III}\) representations of \(\mathcal{O}_{d}\) and corresponding endomorphisms, and the heart of the memoir is a description of an associated family of AF-algebras arising as the fixed-point algebras of the associated modular automorphism groups. Chapters 5–18 are devoted to finding effective methods to decide isomorphism and non-isomorphism in this class of AF-algebras.

#### Readership

Graduate students and research mathematicians interested in functional analysis and operator theory.

#### Table of Contents

# Table of Contents

## Representation Theory and Numerical AF-Invariants: The Representations and Centralizers of Certain States on $\mathcal{O}_{d}$

- Contents v6 free
- Abstract vii8 free
- Preface ix10 free
- Introduction xi12 free
- Part A. Representation Theory 120 free
- Part B. Numerical AF–Invariants 4362
- Chapter 5. The dimension group of u[sub(L)] 4564
- Chapter 6. Invariants related to the Perron–Frobenius eigenvalue 5877
- Chapter 7. The invariants N, D, Prim(m[sub(N)]), Prim(R[sub(D)]), Prim(Q[sub(N–D)]) 6180
- Chapter 8. The invariants K[sub(0)] (u[sub(L)]) [sub(⊗z)]Z[sub(n)] and (ker T)[sub(⊗z)]Z[sub(n)] for n = 2, 3 , 4 , ... 7493
- Chapter 9. Associated structure of the groups K[sub(0)] (u[sub(L)]) and kerT 8099
- Chapter 10. The invariant Ext (T (K[sub(0)] (u[sub(L)]), ker T) 85104
- Chapter 11. Scaling and non–isomorphism 90109
- Chapter 12. Subgroups of G[sub(0)] = U[sup(∞)][sub(n=0)] J[sup(…n)][sub(0)]L 113132
- Chapter 13. Classification of the AF–algebras u[sub(L)] with rank K[sub(0)] (u[sub(L)]) = 2 116135
- Chapter 14. Linear algebra of J 126145
- Chapter 15. Lattice points 129148
- Chapter 16. Complete classification in the cases λ= 2, N = 2, 3,4 131150
- Chapter 17. Complete classification in the case λ = m[sub(N)] 141160
- Chapter 18. Further comments on two examples from Chapter 16 160179

- Bibliography 166185
- List of Figures 170189
- List of Tables 172191
- List of Terms and Symbols 173192